Blood runs thick in Transylvania

Question

Inspector Craig of Scotland Yard was called to Transylvania to solve some cases of vampirism. Arriving there, he found the country inhabited both by vampires and humans. Vampires always lie and humans always tell the truth.

However, half the inhabitants, both human and vampire, are insane and totally deluded in their beliefs: all true propositions they believe false, and all false propositions they believe true. The other half of the inhabitants are completely sane: all true statements they know to be true, and all false statements they know to be false.

Thus sane humans and insane vampires make only true statements; insane humans and sane vampires make only false statements.

Inspector Craig met two sisters, Lucy and Minna. He knew that one was a vampire and one was a human, but knew nothing about the sanity of either. Here is the investigation:

Craig (to Lucy): Tell me about yourselves.

Lucy: We are both insane.

Craig (to Minna): Is that true?

Minna: Of course not!

Did Craig learn the vampire's identity?

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7 answers to this question

If one were sane and the other insane, they would have agreed on that statement. So they are either both sane or both insane. If they are both sane, Lucy would have to be a vampire since she made a false statement. If they are both insane, Lucy would have to be a vampire since she made a true statement. Either way, Lucy is a vampire. Unless Craig is insane, in which case he would be wrong in believing that one is a vampire and one is human